Equivalent expressions
Two expressions are equivalent when they produce the same value for every input. These questions ask you to rewrite an expression — by expanding, factoring, or applying exponent rules — into a form that matches an answer choice.
What College Board tests
Rewriting polynomial and rational expressions in equivalent forms: multiplying out (expanding), factoring, combining like terms, applying exponent rules, and simplifying fractions. The skill is recognizing which operation turns the given form into the one you need.
The core moves
A pattern worth memorizing: the difference of squares, . It appears constantly.
Four worked examples in SAT format. Read the approach, try it yourself, then tap Show the full solution.
1 · Expand a product
Which of the following is equivalent to ?
Approach Multiply every term in the first factor by every term in the second — four products in all — then combine the two middle terms.
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Answer: B
Why the other choices are wrong
A Forgets the middle terms entirely — only multiplies the first and last.
C Sign error: treats the middle as instead of .
D Only keeps the and drops the .
2 · Factor using a pattern
Which of the following is equivalent to ?
Approach Recognize the shape: a perfect square minus a perfect square. is squared and 9 is 3 squared, so this is a difference of squares.
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Answer: B
Check by expanding: . The middle terms cancel — that's the signature of this pattern.
Why the other choices are wrong
A — has a middle term and the wrong constant sign.
C Expands to , not .
D Expands to , not .
3 · Apply exponent rules
Which of the following is equivalent to ?
Approach A power on the outside of a product applies to every factor inside. Raise the 3, the , and the each to the third power.
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Answer: D
Why the other choices are wrong
A Leaves the 3 un-cubed — forgets the coefficient gets the power too.
B Computes as 9 instead of 27.
C Adds the exponents instead of multiplying them.
4 · Simplify a rational expression
For , which of the following is equivalent to ?
Approach Don't divide term by term. Factor the numerator first — it's a difference of squares — then cancel the factor it shares with the denominator.
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Answer: B
Why the other choices are wrong
A Cancels the wrong factor; the in the denominator pairs with the on top, leaving .
C Subtracts the denominator term by term — you can't split a fraction that way.
D Treats the expression as if the variable parts cancel to a constant.